In which cases is the categorical cross-entropy better than the mean squared error?
As a rule of thumb, mean squared error (MSE) is more appropriate for regression problems, that is, problems where the output is a numerical value (i.e. a floating-point number or, in general, a real number).  However, in principle, you can use the MSE for classification problems too (even though that may not be a good idea). MSE can be preceded by the sigmoid function, which outputs a number $p \in [0, 1]$, which can be interpreted as the probability of the input belonging to one of the classes, so the probability of the input belonging to the other class is $1 - p$.

Similarly, cross-entropy (CE) is mainly used for classification problems, that is, problems where the output can belong to one of a discrete set of classes. The CE loss function is usually separately implemented for binary and multi-class classification problems. In the first case, it is called the binary cross-entropy (BCE), and, in the second case, it is called categorical cross-entropy (CCE). The CE requires its inputs to be distributions, so the CCE is usually preceded by a softmax function (so that the resulting vector represents a probability distribution), while the BCE is usually preceded by a sigmoid. 

See also Why is mean squared error the cross-entropy between the empirical distribution and a Gaussian model? for more details about the relationship between the MSE and the cross-entropy. In case you use TensorFlow (TF) or Keras, see also How to choose cross-entropy loss in TensorFlow?, which gives you some guidelines for how to choose the appropriate TF implementation of the cross-entropy function for your (classification) problem. See also Should I use a categorical cross-entropy or binary cross-entropy loss for binary predictions? and Does the cross-entropy cost make sense in the context of regression?.